An artist's style is that 'thing' which makes you recognize a particular painting as being by a particular artist before you’re close enough to see a signature or to read the image label. It can be hard to put into words at times, as it can be quite subtle or understated.
A painting style can be the way the paint is handled (such as Pollock) or the brush strokes (such as Van Gogh). It can be the way a subject is dealt with, or simply the choice of subject(s). It can be the range of colors used, or a particular color that's used in every painting.